# Full text of "Treatise On Analysis Vol-Ii"

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```266        XIV    INTEGRATION IN LOCALLY COMPACT GROUPS

4. HAAR MEASURE ON A QUOTIENT GROUP

Let G be a locally compact group, G' a closed normal subgroup of G,
and G" = G/G' the quotient group, which is locally compact ((12.11.3) and
(12.10.9)). Let a', a" be left Haar measures on G', G", respectively, and let
7i ; G -> G" be the canonical homomorphism.

(14.4.1)    For eachfE Jf (G) and each x e G, the function

= !

JG'

w continuous on G, tf/7^ #(*<!;) = #(A;) /or a// £ 6 G', .yo that we may write
g(x) ~h(n(x)), where h is continuous on G" (12.10.6). The support of h is
compact. The positive linear form

h(x")da"(x")

G"

is a left Haar measure on G.

The continuity of g follows from (14.1.5.5), and we have

because a' is left-invariant. If S = Supp(/), then h(x") ^ 0 implies that
x" e 7r(S), and ;r(S) is compact in G" (3.17.9). Finally, if s is any element of G,
put/! = yCs"1)/, and \Qtg1,hl be the corresponding functions. Then it is clear
that g^(x) = g(sx), and therefore h^x") = h(n(s)xfr). Hence the last part of the
proposition follows from the left-invariance of a" (the fact that the linear form

/»-*• J ^(^") du."(x") is not zero follows from the fact that the supports of a'
and a" are equal to G' and G", respectively).

If we denote by a the left Haar measure on G defined by (14.4.1), then by
abuse of notation we write

(14.4.2)                f f(x) <fa(x) - f  d<f(x) \ f(x

Jo                   JG"          Jo'

t)

where x = TT(JC), and/e

This notational analogy with the product measure (13.21.2) is continued
in the following propositions : of Section 8.5). The number a(A) = a+(A) = a~(A)
```